Orderings in automated theorem proving

This paper is intended to give an introduction to several techniques used in automated theorem proving, and based on the concept of ordering on terms and more generally on formulas. Different orderings and their automation are presented and several applications are considered. We show how orderings provide a way to incorporate equality decision procedures in theorem provers. We also illustrate their fundamental role to automatise proofs by induction and proofs by refutation. Finally, we explain how they allow pruning the search space of a theorem prover by defining strategies to restrict application of inference rules.